
theorem Th46: :: lemma 5.24 (ii), p. 200
  for n being Ordinal, T being connected admissible TermOrder of n
, L being Abelian add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr, P being Subset of Polynom-Ring(n,L), f,g being
  Polynomial of n,L, m being non-zero Monomial of n,L holds f reduces_to g,P,T
  implies m*'f reduces_to m*'g,P,T
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be Abelian
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive almost_left_invertible non degenerated non empty
  doubleLoopStr, P be Subset of Polynom-Ring(n,L), f,g be Polynomial of n,L, m
  be non-zero Monomial of n,L;
  assume f reduces_to g,P,T;
  then consider p being Polynomial of n,L such that
A1: p in P and
A2: f reduces_to g,p,T;
  consider b being bag of n such that
A3: f reduces_to g,p,b,T by A2;
  set b9 = b + HT(m,T);
A4: b in Support f by A3;
A5: now
    m <> 0_(n,L) by POLYNOM7:def 1;
    then Support m <> {} by POLYNOM7:1;
    then
A6: m.term(m) <> 0.L by POLYNOM7:def 5;
    assume
A7: (m*'f).b9 = 0.L;
    (m*'f).b9 = (m*'f).(term(m)+b) by TERMORD:23
      .= m.term(m) * f.b by Th7;
    then f.b = 0.L by A7,A6,VECTSP_2:def 1;
    hence contradiction by A4,POLYNOM1:def 4;
  end;
  b9 is Element of Bags n by PRE_POLY:def 12;
  then
A8: b9 in Support(m*'f) by A5,POLYNOM1:def 4;
A9: p <> 0_(n,L) by A2,Lm18;
  consider s being bag of n such that
A10: s + HT(p,T) = b and
A11: g = f - (f.b/HC(p,T)) * (s *' p) by A3;
  reconsider p as non-zero Polynomial of n,L by A9,POLYNOM7:def 1;
A12: (s + HT(m,T)) + HT(p,T) = b9 by A10,PRE_POLY:35;
  set t = s + HT(m,T);
  set h = (m*'f) - ((m*'f).b9/HC(p,T)) * (t *' p);
  f <> 0_(n,L) by A3;
  then reconsider f as non-zero Polynomial of n,L by POLYNOM7:def 1;
  m*'f <> 0_(n,L) & p <> 0_(n,L) by POLYNOM7:def 1;
  then (m*'f) reduces_to h,p,b9,T by A8,A12;
  then
A13: (m*'f) reduces_to h,p,T;
A14: m.term(m) * (f.b/HC(p,T)) = m.term(m) * (f.b * (HC(p,T)"))
    .= (m.term(m) * f.b) * (HC(p,T)") by GROUP_1:def 3
    .= (m.term(m) * f.b)/HC(p,T);
  (m*'f).b9 = (m*'f).(term(m)+b) by TERMORD:23
    .= m.term(m) * f.b by Th7;
  then h = (m*'f) - (m.term(m) * (f.b/HC(p,T))) * (HT(m,T) *'(s *' p)) by A14
,Th18
    .= (m*'f) - (f.b/HC(p,T)) * (m.term(m) * (HT(m,T) *'(s *' p))) by Th11
    .= (m*'f) - (f.b/HC(p,T)) * (Monom(m.(term(m)),HT(m,T)) *'(s *' p)) by Th22
    .= (m*'f) - (f.b/HC(p,T)) * (Monom(coefficient(m),term(m)) *'(s *' p))
  by TERMORD:23
    .= (m*'f) - (f.b/HC(p,T) * (m *'(s *' p))) by POLYNOM7:11
    .= (m*'f) - (m *' (f.b/HC(p,T) * (s *' p))) by Th12
    .= (m*'f) + -(m *' (f.b/HC(p,T) * (s *' p))) by POLYNOM1:def 7
    .= (m*'f) + (m *' -(f.b/HC(p,T) * (s *' p))) by Th6
    .= m *' (f + -(f.b/HC(p,T)) * (s *' p)) by POLYNOM1:26
    .= m *' (f - (f.b/HC(p,T)) * (s *' p)) by POLYNOM1:def 7;
  hence thesis by A1,A11,A13;
end;
